Quadratic roots, full range

Percentage Accurate: 51.7% → 84.9%

Time: 13.3s

Alternatives: 8

Speedup: 12.9×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 84.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e+73)
   (- (/ c b) (/ b a))
   (if (<= b 6.8e-109)
     (/ (- (sqrt (- (* b b) (* c (* a 4.0)))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+73) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.8e-109) {
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1d+73)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.8d-109) then
        tmp = (sqrt(((b * b) - (c * (a * 4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e+73) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.8e-109) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e+73:
		tmp = (c / b) - (b / a)
	elif b <= 6.8e-109:
		tmp = (math.sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e+73)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.8e-109)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 4.0)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e+73)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.8e-109)
		tmp = (sqrt(((b * b) - (c * (a * 4.0)))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e+73], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-109], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.99999999999999983e72

   1. Initial program 58.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 58.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 58.2%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 94.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 94.2%

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 94.2%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      3. unsub-neg 94.2%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   7. Simplified 94.2%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -9.99999999999999983e72 < b < 6.80000000000000023e-109

   1. Initial program 85.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 6.80000000000000023e-109 < b

   1. Initial program 12.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 12.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 12.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 89.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. mul-1-neg 89.6%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 89.6%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   7. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{+73}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-124)
   (- (/ c b) (/ b a))
   (if (<= b 6.2e-109)
     (* (/ -0.5 a) (- b (sqrt (* a (* c -4.0)))))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-124) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.2e-109) {
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-124)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.2d-109) then
        tmp = ((-0.5d0) / a) * (b - sqrt((a * (c * (-4.0d0)))))
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-124) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.2e-109) {
		tmp = (-0.5 / a) * (b - Math.sqrt((a * (c * -4.0))));
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-124:
		tmp = (c / b) - (b / a)
	elif b <= 6.2e-109:
		tmp = (-0.5 / a) * (b - math.sqrt((a * (c * -4.0))))
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-124)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.2e-109)
		tmp = Float64(Float64(-0.5 / a) * Float64(b - sqrt(Float64(a * Float64(c * -4.0)))));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-124)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.2e-109)
		tmp = (-0.5 / a) * (b - sqrt((a * (c * -4.0))));
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-124], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.2e-109], N[(N[(-0.5 / a), $MachinePrecision] * N[(b - N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.0000000000000003e-124

   1. Initial program 74.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 84.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 84.4%

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 84.4%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      3. unsub-neg 84.4%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   7. Simplified 84.4%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -5.0000000000000003e-124 < b < 6.1999999999999999e-109

   1. Initial program 77.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 77.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 77.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 77.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{a \cdot 2} \]

      2. associate-*r* 77.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]

   7. Simplified 77.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. +-commutative 77.0%

         \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} + \left(-b\right)}}{a \cdot 2} \]

      2. unsub-neg 77.0%

         \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}{a \cdot 2} \]

   9. Applied egg-rr 77.0%

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}{a \cdot 2} \]
   10. Step-by-step derivation

       1. frac-2neg 77.0%

          \[\leadsto \color{blue}{\frac{-\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)}{-a \cdot 2}} \]

       2. div-inv 77.0%

          \[\leadsto \color{blue}{\left(-\left(\sqrt{a \cdot \left(c \cdot -4\right)} - b\right)\right) \cdot \frac{1}{-a \cdot 2}} \]

   11. Applied egg-rr 77.0%

       \[\leadsto \color{blue}{\left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right) \cdot \frac{-0.5}{a}} \]
   12. Step-by-step derivation

       1. *-commutative 77.0%

          \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

   13. Simplified 77.0%

       \[\leadsto \color{blue}{\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)} \]

   if 6.1999999999999999e-109 < b

   1. Initial program 12.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 12.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 12.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 89.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. mul-1-neg 89.6%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 89.6%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   7. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-124}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{-0.5}{a} \cdot \left(b - \sqrt{a \cdot \left(c \cdot -4\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-124}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.16e-124)
   (- (/ c b) (/ b a))
   (if (<= b 6.6e-109)
     (/ (- (sqrt (* a (* c -4.0))) b) (* a 2.0))
     (/ c (- b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-124) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6e-109) {
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-1.16d-124)) then
        tmp = (c / b) - (b / a)
    else if (b <= 6.6d-109) then
        tmp = (sqrt((a * (c * (-4.0d0)))) - b) / (a * 2.0d0)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.16e-124) {
		tmp = (c / b) - (b / a);
	} else if (b <= 6.6e-109) {
		tmp = (Math.sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.16e-124:
		tmp = (c / b) - (b / a)
	elif b <= 6.6e-109:
		tmp = (math.sqrt((a * (c * -4.0))) - b) / (a * 2.0)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.16e-124)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	elseif (b <= 6.6e-109)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -4.0))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.16e-124)
		tmp = (c / b) - (b / a);
	elseif (b <= 6.6e-109)
		tmp = (sqrt((a * (c * -4.0))) - b) / (a * 2.0);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.16e-124], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.6e-109], N[(N[(N[Sqrt[N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.1600000000000001e-124

   1. Initial program 74.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 84.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 84.4%

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 84.4%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      3. unsub-neg 84.4%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   7. Simplified 84.4%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -1.1600000000000001e-124 < b < 6.59999999999999981e-109

   1. Initial program 77.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 77.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 77.1%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around 0 77.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-4 \cdot \left(a \cdot c\right)}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. *-commutative 77.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -4}}}{a \cdot 2} \]

      2. associate-*r* 77.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]

   7. Simplified 77.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(c \cdot -4\right)}}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. +-commutative 77.0%

         \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} + \left(-b\right)}}{a \cdot 2} \]

      2. unsub-neg 77.0%

         \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}{a \cdot 2} \]

   9. Applied egg-rr 77.0%

      \[\leadsto \frac{\color{blue}{\sqrt{a \cdot \left(c \cdot -4\right)} - b}}{a \cdot 2} \]

   if 6.59999999999999981e-109 < b

   1. Initial program 12.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 12.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 12.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 89.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. mul-1-neg 89.6%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 89.6%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   7. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.16 \cdot 10^{-124}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{elif}\;b \leq 6.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -4\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (- (/ c b) (/ b a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (c / b) - (b / a)
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (c / b) - (b / a);
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (c / b) - (b / a)
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(c / b) - Float64(b / a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (c / b) - (b / a);
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(c / b), $MachinePrecision] - N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 70.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a} + \frac{c}{b}} \]
   6. Step-by-step derivation

      1. +-commutative 70.7%

         \[\leadsto \color{blue}{\frac{c}{b} + -1 \cdot \frac{b}{a}} \]

      2. mul-1-neg 70.7%

         \[\leadsto \frac{c}{b} + \color{blue}{\left(-\frac{b}{a}\right)} \]

      3. unsub-neg 70.7%

         \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   7. Simplified 70.7%

      \[\leadsto \color{blue}{\frac{c}{b} - \frac{b}{a}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 31.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 31.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 65.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. mul-1-neg 65.6%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 65.6%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   7. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{c}{b} - \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 42.9% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (if (<= b 1.1e+31) (/ b (- a)) (/ c b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1e+31) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.1d+31) then
        tmp = b / -a
    else
        tmp = c / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.1e+31) {
		tmp = b / -a;
	} else {
		tmp = c / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.1e+31:
		tmp = b / -a
	else:
		tmp = c / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.1e+31)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.1e+31)
		tmp = b / -a;
	else
		tmp = c / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.1e+31], N[(b / (-a)), $MachinePrecision], N[(c / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.10000000000000005e31

   1. Initial program 70.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 70.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 49.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 49.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg 49.6%

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   7. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if 1.10000000000000005e31 < b

   1. Initial program 8.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 8.5%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 8.5%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 86.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
   6. Step-by-step derivation

      1. associate-/l* 88.3%

         \[\leadsto \frac{-2 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}}{a \cdot 2} \]

   7. Simplified 88.3%

      \[\leadsto \frac{\color{blue}{-2 \cdot \left(a \cdot \frac{c}{b}\right)}}{a \cdot 2} \]
   8. Step-by-step derivation

      1. div-inv 88.2%

         \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot \frac{c}{b}\right)\right) \cdot \frac{1}{a \cdot 2}} \]

      2. associate-*r* 88.3%

         \[\leadsto \color{blue}{\left(\left(-2 \cdot a\right) \cdot \frac{c}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

      3. metadata-eval 88.3%

         \[\leadsto \left(\left(\color{blue}{\left(-2\right)} \cdot a\right) \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

      4. distribute-lft-neg-in 88.3%

         \[\leadsto \left(\color{blue}{\left(-2 \cdot a\right)} \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

      5. *-commutative 88.3%

         \[\leadsto \left(\left(-\color{blue}{a \cdot 2}\right) \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

      6. frac-2neg 88.3%

         \[\leadsto \left(\left(-a \cdot 2\right) \cdot \color{blue}{\frac{-c}{-b}}\right) \cdot \frac{1}{a \cdot 2} \]

      7. distribute-frac-neg 88.3%

         \[\leadsto \left(\left(-a \cdot 2\right) \cdot \color{blue}{\left(-\frac{c}{-b}\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      8. add-sqr-sqrt 0.0%

         \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      9. sqrt-unprod 38.0%

         \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      10. sqr-neg 38.0%

          \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\sqrt{\color{blue}{b \cdot b}}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      11. sqrt-prod 37.5%

          \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      12. add-sqr-sqrt 37.5%

          \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\color{blue}{b}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      13. distribute-rgt-neg-in 37.5%

          \[\leadsto \color{blue}{\left(-\left(-a \cdot 2\right) \cdot \frac{c}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

      14. *-commutative 37.5%

          \[\leadsto \left(-\left(-\color{blue}{2 \cdot a}\right) \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

      15. distribute-lft-neg-in 37.5%

          \[\leadsto \left(-\color{blue}{\left(\left(-2\right) \cdot a\right)} \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

      16. metadata-eval 37.5%

          \[\leadsto \left(-\left(\color{blue}{-2} \cdot a\right) \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

      17. associate-*r* 37.5%

          \[\leadsto \left(-\color{blue}{-2 \cdot \left(a \cdot \frac{c}{b}\right)}\right) \cdot \frac{1}{a \cdot 2} \]

      18. distribute-lft-neg-in 37.5%

          \[\leadsto \color{blue}{\left(\left(--2\right) \cdot \left(a \cdot \frac{c}{b}\right)\right)} \cdot \frac{1}{a \cdot 2} \]

      19. metadata-eval 37.5%

          \[\leadsto \left(\color{blue}{2} \cdot \left(a \cdot \frac{c}{b}\right)\right) \cdot \frac{1}{a \cdot 2} \]

      20. associate-*r/ 37.5%

          \[\leadsto \left(2 \cdot \color{blue}{\frac{a \cdot c}{b}}\right) \cdot \frac{1}{a \cdot 2} \]

      21. metadata-eval 37.5%

          \[\leadsto \left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{0.5}}} \]

      22. div-inv 37.5%

          \[\leadsto \left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{1}{\color{blue}{\frac{a}{0.5}}} \]

   9. Applied egg-rr 37.5%

      \[\leadsto \color{blue}{\left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{0.5}{a}} \]
   10. Step-by-step derivation

       1. associate-*r/ 37.5%

          \[\leadsto \color{blue}{\frac{\left(2 \cdot \frac{a \cdot c}{b}\right) \cdot 0.5}{a}} \]

       2. *-commutative 37.5%

          \[\leadsto \frac{\color{blue}{\left(\frac{a \cdot c}{b} \cdot 2\right)} \cdot 0.5}{a} \]

       3. associate-*l* 37.5%

          \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} \cdot \left(2 \cdot 0.5\right)}}{a} \]

       4. metadata-eval 37.5%

          \[\leadsto \frac{\frac{a \cdot c}{b} \cdot \color{blue}{1}}{a} \]

       5. associate-/l* 37.5%

          \[\leadsto \frac{\color{blue}{\left(a \cdot \frac{c}{b}\right)} \cdot 1}{a} \]

   11. Simplified 37.5%

       \[\leadsto \color{blue}{\frac{\left(a \cdot \frac{c}{b}\right) \cdot 1}{a}} \]

   12. Taylor expanded in a around 0 37.3%

       \[\leadsto \color{blue}{\frac{c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.5% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (- a)) (/ c (- b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / -a
    else
        tmp = c / -b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / -a;
	} else {
		tmp = c / -b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / -a
	else:
		tmp = c / -b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(-a));
	else
		tmp = Float64(c / Float64(-b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / -a;
	else
		tmp = c / -b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / (-a)), $MachinePrecision], N[(c / (-b)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.6%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 77.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around -inf 70.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 70.6%

         \[\leadsto \color{blue}{\frac{-1 \cdot b}{a}} \]

      2. mul-1-neg 70.6%

         \[\leadsto \frac{\color{blue}{-b}}{a} \]

   7. Simplified 70.6%

      \[\leadsto \color{blue}{\frac{-b}{a}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 31.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 31.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 31.3%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing

   5. Taylor expanded in b around inf 65.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{c}{b}} \]
   6. Step-by-step derivation

      1. mul-1-neg 65.6%

         \[\leadsto \color{blue}{-\frac{c}{b}} \]

      2. distribute-neg-frac 65.6%

         \[\leadsto \color{blue}{\frac{-c}{b}} \]

   7. Simplified 65.6%

      \[\leadsto \color{blue}{\frac{-c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{-a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{-b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 2.5% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b a))

C

double code(double a, double b, double c) {
	return b / a;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / a
end function

Java

public static double code(double a, double b, double c) {
	return b / a;
}

Python

def code(a, b, c):
	return b / a

Julia

function code(a, b, c)
	return Float64(b / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / a;
end

Wolfram

code[a_, b_, c_] := N[(b / a), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 55.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 55.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - b}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. *-un-lft-identity 55.4%

      \[\leadsto \frac{\color{blue}{1 \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} - b}{a \cdot 2} \]

   2. *-un-lft-identity 55.4%

      \[\leadsto \frac{1 \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} - \color{blue}{1 \cdot b}}{a \cdot 2} \]

   3. prod-diff 55.4%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1, \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, -b \cdot 1\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}}{a \cdot 2} \]

   4. *-commutative 55.4%

      \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, -\color{blue}{1 \cdot b}\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   5. *-un-lft-identity 55.4%

      \[\leadsto \frac{\mathsf{fma}\left(1, \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}, -\color{blue}{b}\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   6. fma-define 55.4%

      \[\leadsto \frac{\color{blue}{\left(1 \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)} + \left(-b\right)\right)} + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   7. *-un-lft-identity 55.4%

      \[\leadsto \frac{\left(\color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}} + \left(-b\right)\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   8. +-commutative 55.4%

      \[\leadsto \frac{\color{blue}{\left(\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right)} + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   9. add-sqr-sqrt 40.3%

      \[\leadsto \frac{\left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   10. sqrt-unprod 53.3%

       \[\leadsto \frac{\left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   11. sqr-neg 53.3%

       \[\leadsto \frac{\left(\sqrt{\color{blue}{b \cdot b}} + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   12. sqrt-prod 13.0%

       \[\leadsto \frac{\left(\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   13. add-sqr-sqrt 32.8%

       \[\leadsto \frac{\left(\color{blue}{b} + \sqrt{\mathsf{fma}\left(a, c \cdot -4, b \cdot b\right)}\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   14. pow2 32.8%

       \[\leadsto \frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, \color{blue}{{b}^{2}}\right)}\right) + \mathsf{fma}\left(-b, 1, b \cdot 1\right)}{a \cdot 2} \]

   15. add-sqr-sqrt 20.5%

       \[\leadsto \frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}, 1, b \cdot 1\right)}{a \cdot 2} \]

   16. sqrt-unprod 32.8%

       \[\leadsto \frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}, 1, b \cdot 1\right)}{a \cdot 2} \]

   17. sqr-neg 32.8%

       \[\leadsto \frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(\sqrt{\color{blue}{b \cdot b}}, 1, b \cdot 1\right)}{a \cdot 2} \]

   18. sqrt-prod 13.0%

       \[\leadsto \frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(\color{blue}{\sqrt{b} \cdot \sqrt{b}}, 1, b \cdot 1\right)}{a \cdot 2} \]

   19. add-sqr-sqrt 32.4%

       \[\leadsto \frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(\color{blue}{b}, 1, b \cdot 1\right)}{a \cdot 2} \]

   20. *-commutative 32.4%

       \[\leadsto \frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(b, 1, \color{blue}{1 \cdot b}\right)}{a \cdot 2} \]

   21. *-un-lft-identity 32.4%

       \[\leadsto \frac{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(b, 1, \color{blue}{b}\right)}{a \cdot 2} \]

6. Applied egg-rr 32.4%

   \[\leadsto \frac{\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)}\right) + \mathsf{fma}\left(b, 1, b\right)}}{a \cdot 2} \]
7. Step-by-step derivation

   1. +-commutative 32.4%

      \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} + b\right)} + \mathsf{fma}\left(b, 1, b\right)}{a \cdot 2} \]

   2. associate-+l+ 32.4%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} + \left(b + \mathsf{fma}\left(b, 1, b\right)\right)}}{a \cdot 2} \]

   3. fma-undefine 32.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} + \left(b + \color{blue}{\left(b \cdot 1 + b\right)}\right)}{a \cdot 2} \]

   4. *-rgt-identity 32.4%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} + \left(b + \left(\color{blue}{b} + b\right)\right)}{a \cdot 2} \]

8. Simplified 32.4%

   \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -4, {b}^{2}\right)} + \left(b + \left(b + b\right)\right)}}{a \cdot 2} \]

9. Taylor expanded in b around -inf 2.4%

   \[\leadsto \color{blue}{\frac{b}{a}} \]

10. Final simplification 2.4%

    \[\leadsto \frac{b}{a} \]
11. Add Preprocessing

Alternative 8: 10.8% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ c b))

C

double code(double a, double b, double c) {
	return c / b;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = c / b
end function

Java

public static double code(double a, double b, double c) {
	return c / b;
}

Python

def code(a, b, c):
	return c / b

Julia

function code(a, b, c)
	return Float64(c / b)
end

MATLAB

function tmp = code(a, b, c)
	tmp = c / b;
end

Wolfram

code[a_, b_, c_] := N[(c / b), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 55.3%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 55.3%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 26.9%

   \[\leadsto \frac{\color{blue}{-2 \cdot \frac{a \cdot c}{b}}}{a \cdot 2} \]
6. Step-by-step derivation

   1. associate-/l* 28.8%

      \[\leadsto \frac{-2 \cdot \color{blue}{\left(a \cdot \frac{c}{b}\right)}}{a \cdot 2} \]

7. Simplified 28.8%

   \[\leadsto \frac{\color{blue}{-2 \cdot \left(a \cdot \frac{c}{b}\right)}}{a \cdot 2} \]
8. Step-by-step derivation

   1. div-inv 28.7%

      \[\leadsto \color{blue}{\left(-2 \cdot \left(a \cdot \frac{c}{b}\right)\right) \cdot \frac{1}{a \cdot 2}} \]

   2. associate-*r* 28.8%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot a\right) \cdot \frac{c}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

   3. metadata-eval 28.8%

      \[\leadsto \left(\left(\color{blue}{\left(-2\right)} \cdot a\right) \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

   4. distribute-lft-neg-in 28.8%

      \[\leadsto \left(\color{blue}{\left(-2 \cdot a\right)} \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

   5. *-commutative 28.8%

      \[\leadsto \left(\left(-\color{blue}{a \cdot 2}\right) \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

   6. frac-2neg 28.8%

      \[\leadsto \left(\left(-a \cdot 2\right) \cdot \color{blue}{\frac{-c}{-b}}\right) \cdot \frac{1}{a \cdot 2} \]

   7. distribute-frac-neg 28.8%

      \[\leadsto \left(\left(-a \cdot 2\right) \cdot \color{blue}{\left(-\frac{c}{-b}\right)}\right) \cdot \frac{1}{a \cdot 2} \]

   8. add-sqr-sqrt 1.2%

      \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

   9. sqrt-unprod 10.9%

      \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

   10. sqr-neg 10.9%

       \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\sqrt{\color{blue}{b \cdot b}}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

   11. sqrt-prod 9.6%

       \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

   12. add-sqr-sqrt 11.6%

       \[\leadsto \left(\left(-a \cdot 2\right) \cdot \left(-\frac{c}{\color{blue}{b}}\right)\right) \cdot \frac{1}{a \cdot 2} \]

   13. distribute-rgt-neg-in 11.6%

       \[\leadsto \color{blue}{\left(-\left(-a \cdot 2\right) \cdot \frac{c}{b}\right)} \cdot \frac{1}{a \cdot 2} \]

   14. *-commutative 11.6%

       \[\leadsto \left(-\left(-\color{blue}{2 \cdot a}\right) \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

   15. distribute-lft-neg-in 11.6%

       \[\leadsto \left(-\color{blue}{\left(\left(-2\right) \cdot a\right)} \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

   16. metadata-eval 11.6%

       \[\leadsto \left(-\left(\color{blue}{-2} \cdot a\right) \cdot \frac{c}{b}\right) \cdot \frac{1}{a \cdot 2} \]

   17. associate-*r* 11.6%

       \[\leadsto \left(-\color{blue}{-2 \cdot \left(a \cdot \frac{c}{b}\right)}\right) \cdot \frac{1}{a \cdot 2} \]

   18. distribute-lft-neg-in 11.6%

       \[\leadsto \color{blue}{\left(\left(--2\right) \cdot \left(a \cdot \frac{c}{b}\right)\right)} \cdot \frac{1}{a \cdot 2} \]

   19. metadata-eval 11.6%

       \[\leadsto \left(\color{blue}{2} \cdot \left(a \cdot \frac{c}{b}\right)\right) \cdot \frac{1}{a \cdot 2} \]

   20. associate-*r/ 11.6%

       \[\leadsto \left(2 \cdot \color{blue}{\frac{a \cdot c}{b}}\right) \cdot \frac{1}{a \cdot 2} \]

   21. metadata-eval 11.6%

       \[\leadsto \left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{1}{a \cdot \color{blue}{\frac{1}{0.5}}} \]

   22. div-inv 11.6%

       \[\leadsto \left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{1}{\color{blue}{\frac{a}{0.5}}} \]

9. Applied egg-rr 11.6%

   \[\leadsto \color{blue}{\left(2 \cdot \frac{a \cdot c}{b}\right) \cdot \frac{0.5}{a}} \]
10. Step-by-step derivation

    1. associate-*r/ 11.6%

       \[\leadsto \color{blue}{\frac{\left(2 \cdot \frac{a \cdot c}{b}\right) \cdot 0.5}{a}} \]

    2. *-commutative 11.6%

       \[\leadsto \frac{\color{blue}{\left(\frac{a \cdot c}{b} \cdot 2\right)} \cdot 0.5}{a} \]

    3. associate-*l* 11.6%

       \[\leadsto \frac{\color{blue}{\frac{a \cdot c}{b} \cdot \left(2 \cdot 0.5\right)}}{a} \]

    4. metadata-eval 11.6%

       \[\leadsto \frac{\frac{a \cdot c}{b} \cdot \color{blue}{1}}{a} \]

    5. associate-/l* 11.6%

       \[\leadsto \frac{\color{blue}{\left(a \cdot \frac{c}{b}\right)} \cdot 1}{a} \]

11. Simplified 11.6%

    \[\leadsto \color{blue}{\frac{\left(a \cdot \frac{c}{b}\right) \cdot 1}{a}} \]

12. Taylor expanded in a around 0 11.6%

    \[\leadsto \color{blue}{\frac{c}{b}} \]

13. Final simplification 11.6%

    \[\leadsto \frac{c}{b} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (a b c)
  :name "Quadratic roots, full range"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))